such that it performs the following operations given an initial mean-field interaction $\hat{V}_{\text{init, MF}}$:
which is defined in {autolink}`~pymf.model.Model.mfield` method.
It performs the following steps:
1. Calculate the total Hamiltonian $\hat{H}(R) = \hat{H_0}(R) + \hat{V}_{\text{init, MF}}(R)$ in real-space.
2.Fourier transform the total Hamiltonian to the momentum space $\hat{H}(R) \to \hat{H}(k)$.
3. Calculate the ground-state density matrix $\rho_{mn}(k)$ in momentum space.
1. Diagonalize the Hamiltonian $\hat{H}(k)$ to obtain the eigenvalues and eigenvectors.
2. Calculate the fermi level $\mu$ given the desired filling of the unit cell.
3. Calculate the density matrix $\rho_{mn}(k)$ using the eigenvectors and the fermi level $\mu$ (currently we do not consider thermal effects so $\beta \to \infty$).
4. Inverse Fourier transform the density matrix to real-space $\rho_{mn}(k) \to \rho_{mn}(R)$.
5. Calculate the new mean-field interaction $\hat{V}_{\text{new, MF}}(R)$ via {eq}`mf_infinite`.
2.({autolink}`~pymf.mf.construct_density_matrix`) Compute the ground-state density matrix $\rho_{mn}(R)$:
1. ({autolink}`~pymf.tb.transforms.tb_to_kgrid`) Fourier transform the total Hamiltonian to momentum space $\hat{H}(R) \to \hat{H}(k)$.
2. ({autolink}`numpy.linalg.eigh`) Diagonalize the Hamiltonian $\hat{H}(R)$ to obtain the eigenvalues and eigenvectors.
3.({autolink}`~pymf.mf.fermi_on_kgrid`) Calculate the fermi level given the desired filling of the unit cell.
4. ({autolink}`~pymf.mf.construct_density_matrix_kgrid`) Calculate the density matrix $\rho_{mn}(k)$ using the eigenvectors and the fermi level.
5. ({autolink}`~pymf.tb.transforms.kgrid_to_tb`) Inverse Fourier transform the density matrix to real-space $\rho_{mn}(k) \to \rho_{mn}(R)$.
3.({autolink}`~pymf.mf.meanfield`) Calculate the new mean-field correction $\hat{V}_{\text{new, MF}}(R)$ using {eq}`mf_infinite`.
## Self-consistency criterion
## Self-consistency criteria
To define the self-consistency condition, we first introduce an invertible function $f$ that uniquely maps $\hat{V}_{\text{MF}}$ to a real-valued vector which minimally parameterizes it:
...
...
@@ -43,13 +43,15 @@ $$
f : \hat{V}_{\text{MF}} \to f(\hat{V}_{\text{MF}}) \in \mathbb{R}^N.
$$
In the code, $f$ corresponds to the {autolink}`~pymf.params.rparams.tb_to_rparams` function (inverse is {autolink}`~pymf.params.rparams.rparams_to_tb`).
Currently, $f$ parameterizes the mean-field interaction by taking only the upper triangular elements of the matrix $V_{\text{MF}, nm}(R)$ (the lower triangular part is redundant due to the Hermiticity of the Hamiltonian) and splitting it into a real and imaginary parts to form a real-valued vector.
With this function, we define the self-consistency criterion as a fixed-point problem:
With this, we define the self-consistency criterion as a fixed-point problem:
To solve this fixed-point problem, we utilize a root-finding function `scipy.optimize.anderson` which uses the Anderson mixing method to find the fixed-point solution.
However, our implementation also allows to use other custom fixed-point solvers by either providing it to `solver` or by re-defining the `solver` function.
Instead of solving the fixed point problem, we rewrite is as the difference of the two successive self-consistent mean-field iterations in {autolink}`~pymf.solvers.cost`.
That re-defines the problem into a root-finding problem which is more consistent with available numerical solvers such as {autolink}`~scipy.optimize.anderson`.
That is exactly what we do in the {autolink}`~pymf.solvers.solver` function, although we also provide the option to use a custom optimizer.
@@ -31,24 +31,25 @@ Such a task is often infeasible for large systems and one needs to resort to app
## Mean-field approximaton
The first-order perturbative approximation to the interacting Hamiltonian is the Hartree-Fock approximation also known as the mean-field approximation.
The mean-field approximates the quartic term $\hat{V}$ in {eq}`hamiltonian` as a sum of bilinear terms weighted by the expectation values the remaining operators:
The mean-field approximates the quartic term $\hat{V}$ in {eq}`hamiltonian` as a sum of bilinear terms weighted by the expectation values of the remaining operators:
we neglect the superconducting pairing and constant offset terms.
The expectation value terms $\langle c_i^\dagger c_j \rangle$ are due to the ground-state density matrix and therefore act as an effective field acting on the system.
The ground-state density matrix reads:
where we neglect the constant offset terms and the superconducting pairing (for now).
The expectation value terms $\langle c_i^\dagger c_j \rangle$ are due to the groundstate density matrix and act as an effective field on the system.
where $\beta = 1/ (k_B T)$ is the inverse temperature, $\mu$ is the chemical potential, and $\hat{N} = \sum_i c_i^\dagger c_i$ is the number operator.
Currently, we neglect thermal effects and so $\beta \to \infty$.
## Finite tight-binding grid
To simplify the mean-field Hamiltonian, we assume a finite, normalised orthogonal tight-binding grid defined by the single-particle basis states:
To simplify the mean-field Hamiltonian, we assume a finite, normalised, orthogonal tight-binding grid defined by the single-particle basis states:
$$
\ket{n} = c^\dagger_n\ket{\text{vac}}
...
...
@@ -71,7 +72,7 @@ $$
n \to n, R_n.
$$
Because of the translationaly invariance, the physical properties of the system are independent of the absolute unit cell position $R_n$ and rather depend on the relative position between the two unit cells $R_{nm} = R_n - R_m$:
Because of the translational invariance, the physical properties of the system are independent of the absolute unit cell position $R_n$, but rather depend on the relative position between the two unit cells $R_{nm} = R_n - R_m$:
